Abstract
An induced matching of a graph G = (V,E) is a matching \({\mathcal M}\) such that no two edges of \({\mathcal M}\) are joined by an edge of E/\({\mathcal M}\) In general, the problem of finding a maximum induced matching of a graph is known to be NP-hard. In random d-regular graphs, the problem of finding a maximum induced matching has been studied for d ∈ {3, 4, ..., 10 }. This was due to Duckworth et al.(2002) where they gave the asymptotically almost sure lower bounds and upper bonds on the size of maximum induced matchings in such graphs. The asymptotically almost sure lower bounds were achieved by analysing a degree-greedy algorithm using the differential equation method, whilst the asymptotically almost sure upper bounds were obtained by a direct expectation argument. In this paper, using the small subgraph conditioning method, we will show the asymptotically almost sure existence of an induced matching of certain size in random d-regular graphs, for d ∈ {3,4, 5}. This result improves the known asymptotically almost sure lower bound obtained by Duckworth et al.(2002).
The research was carried out while the author was in the Department of Mathematics & Statistics, the University of Melbourne, Australia.
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References
Assiyatun, H.: Large Subgraphs of Regular Graphs, Doctoral Thesis, Department of Mathematics and Statistics, The University of Melbourne, Australia (2001)
Assiyatun, H., Duckworth, W.: Small Maximal Matchings of Random Cubic Graph (preprint)
Beis, M., Duckworth, W., Zito, M.: Packing Edges in Random Regular Graphs. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 118–130. Springer, Heidelberg (2002)
Bollobás, B.: Random Graphs. In: Temperley, H.N.V. (ed.) Combinatorics. London Mathematical Society Lecture Note Series, vol. 52, pp. 80–102. Cambridge University Press, Cambridge (1981)
Cameron, K.: Induced Matchings. Discrete Applied Mathematics 24, 97–102 (1989)
Duckworth, W., Wormald, N.C., Zito, M.: Maximum Induced Matchings of Random Cubic Graph. The Journal of Computational and Applied Mathematics 142(1), 39–50 (2002)
Duckworth, W., Manlove, D., Zito, M.: On the Approximability of the Maximum Induced Matching Problem, Technical Report, TR-2000-56, Department of Computing Science of Glasgow University (2000)
Garmo, H.: Random Railways and Cycles in Random Regular Graphs, Doctoral Thesis, Uppsala University, Sweden (1998)
Golumbic, M.C., Lewenstein, M.: New results on induced matchings. Discrete Applied Mathematics 101, 157–165 (2000)
Golumbic, M.C., Laskar, R.C.: Irredundancy in Circular Arc Graphs. Discrete Applied Mathematics 44, 79–89 (1993)
Robalewska, H.D.: 2-Factors in Random Regular Graphs. Journal of Graph Theory 23(3), 215–224 (1996)
Janson, S.: Random Regular Graphs: Asymptotic Distributions and Contiguity. Combinatorics, Probability and Computing 4(4), 369–405 (1995)
Robinson, R.W., Wormald, N.C.: Almost All Regular Graphs are Hamiltonian. Random Structures and Algorithms 5(2), 363–374 (1994)
Robinson, R.W., Wormald, N.C.: Almost All Cubic Graphs are Hamiltonian. Random Structures & Algorithms 3, 117–125 (1992)
Stockmeyer, L.J., Vazirani, V.V.: NP-Completeness of Some Generalizations of the Maximum Matching Problem. Information Processing Letters 15(1), 14–19 (1982)
Wormald, N.C.: Models of Random Regular Graphs. In: Surveys in Combinatorics, pp. 239–298. Cambridge University Press, Cambridge (Canterbury 1999)
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Assiyatun, H. (2005). Maximum Induced Matchings of Random Regular Graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds) Combinatorial Geometry and Graph Theory. IJCCGGT 2003. Lecture Notes in Computer Science, vol 3330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30540-8_5
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DOI: https://doi.org/10.1007/978-3-540-30540-8_5
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